diff --git a/src/extension/op_def.rs b/src/extension/op_def.rs
index 9dc2fcd18..d1062a3ce 100644
--- a/src/extension/op_def.rs
+++ b/src/extension/op_def.rs
@@ -180,9 +180,6 @@ impl OpDef {
         args: &[TypeArg],
         exts: &ExtensionRegistry,
     ) -> Result<FunctionType, SignatureError> {
-        // Hugr's are monomorphic, so check the args have no free variables
-        args.iter().try_for_each(|ta| ta.validate(exts, &[]))?;
-
         let temp: PolyFuncType; // to keep alive
         let (pf, args) = match &self.signature_func {
             SignatureFunc::TypeScheme(ts) => (ts, args),
@@ -369,11 +366,16 @@ impl Extension {
 
 #[cfg(test)]
 mod test {
+    use std::num::NonZeroU64;
+
     use smol_str::SmolStr;
 
     use crate::builder::{DFGBuilder, Dataflow, DataflowHugr};
     use crate::extension::prelude::USIZE_T;
-    use crate::extension::{ExtensionRegistry, PRELUDE};
+    use crate::extension::{
+        CustomSignatureFunc, ExtensionRegistry, SignatureError, EMPTY_REG, PRELUDE,
+        PRELUDE_REGISTRY,
+    };
     use crate::ops::custom::ExternalOp;
     use crate::ops::LeafOp;
     use crate::std_extensions::collections::{EXTENSION, LIST_TYPENAME};
@@ -413,4 +415,119 @@ mod test {
 
         Ok(())
     }
+
+    #[test]
+    fn binary_polyfunc() -> Result<(), Box<dyn std::error::Error>> {
+        // Test a custom binary `compute_signature` that returns a PolyFuncType
+        // where the latter declares more type params itself. In particular,
+        // we should be able to substitute (external) type variables into the latter,
+        // but not pass them into the former (custom binary function).
+        struct SigFun();
+        impl CustomSignatureFunc for SigFun {
+            fn compute_signature(
+                &self,
+                _name: &SmolStr,
+                arg_values: &[TypeArg],
+                _misc: &std::collections::HashMap<String, serde_yaml::Value>,
+                _exts: &ExtensionRegistry,
+            ) -> Result<PolyFuncType, SignatureError> {
+                const TP: TypeParam = TypeParam::Type(TypeBound::Any);
+                let [TypeArg::BoundedNat { n }] = arg_values else {
+                    return Err(SignatureError::InvalidTypeArgs);
+                };
+                let n = *n as usize;
+                let tvs: Vec<Type> = (0..n)
+                    .map(|_| Type::new_var_use(0, TypeBound::Any))
+                    .collect();
+                Ok(PolyFuncType::new(
+                    vec![TP],
+                    FunctionType::new(tvs.clone(), vec![Type::new_tuple(tvs)]),
+                ))
+            }
+        }
+        let mut e = Extension::new(EXT_ID);
+        let def = e.add_op_custom_sig_simple(
+            "MyOp".into(),
+            "".to_string(),
+            vec![TypeParam::max_nat()],
+            SigFun(),
+        )?;
+
+        // Base case, no type variables:
+        let args = [TypeArg::BoundedNat { n: 3 }, USIZE_T.into()];
+        assert_eq!(
+            def.compute_signature(&args, &PRELUDE_REGISTRY),
+            Ok(FunctionType::new(
+                vec![USIZE_T; 3],
+                vec![Type::new_tuple(vec![USIZE_T; 3])]
+            ))
+        );
+        assert_eq!(def.validate_args(&args, &PRELUDE_REGISTRY, &[]), Ok(()));
+
+        // Second arg may be a variable (substitutable)
+        let tyvar = Type::new_var_use(0, TypeBound::Eq);
+        let tyvars: Vec<Type> = vec![tyvar.clone(); 3];
+        let args = [TypeArg::BoundedNat { n: 3 }, tyvar.clone().into()];
+        assert_eq!(
+            def.compute_signature(&args, &PRELUDE_REGISTRY),
+            Ok(FunctionType::new(
+                tyvars.clone(),
+                vec![Type::new_tuple(tyvars)]
+            ))
+        );
+        def.validate_args(&args, &PRELUDE_REGISTRY, &[TypeParam::Type(TypeBound::Eq)])
+            .unwrap();
+
+        // quick sanity check that we are validating the args - note changed bound:
+        assert_eq!(
+            def.validate_args(&args, &PRELUDE_REGISTRY, &[TypeParam::Type(TypeBound::Any)]),
+            Err(SignatureError::TypeVarDoesNotMatchDeclaration {
+                actual: TypeBound::Any.into(),
+                cached: TypeBound::Eq.into()
+            })
+        );
+
+        // First arg must be concrete, not a variable
+        let kind = TypeParam::bounded_nat(NonZeroU64::new(5).unwrap());
+        let args = [TypeArg::new_var_use(0, kind.clone()), USIZE_T.into()];
+        // We can't prevent this from getting into our compute_signature implementation:
+        assert_eq!(
+            def.compute_signature(&args, &PRELUDE_REGISTRY),
+            Err(SignatureError::InvalidTypeArgs)
+        );
+        // But validation rules it out, even when the variable is declared:
+        assert_eq!(
+            def.validate_args(&args, &PRELUDE_REGISTRY, &[kind]),
+            Err(SignatureError::FreeTypeVar {
+                idx: 0,
+                num_decls: 0
+            })
+        );
+
+        Ok(())
+    }
+
+    #[test]
+    fn type_scheme_instantiate_var() -> Result<(), Box<dyn std::error::Error>> {
+        // Check that we can instantiate a PolyFuncType-scheme with an (external)
+        // type variable
+        let mut e = Extension::new(EXT_ID);
+        let def = e.add_op_type_scheme_simple(
+            "SimpleOp".into(),
+            "".into(),
+            PolyFuncType::new(
+                vec![TypeParam::Type(TypeBound::Any)],
+                FunctionType::new_endo(vec![Type::new_var_use(0, TypeBound::Any)]),
+            ),
+        )?;
+        let tv = Type::new_var_use(1, TypeBound::Eq);
+        let args = [TypeArg::Type { ty: tv.clone() }];
+        let decls = [TypeParam::Extensions, TypeBound::Eq.into()];
+        def.validate_args(&args, &EMPTY_REG, &decls).unwrap();
+        assert_eq!(
+            def.compute_signature(&args, &EMPTY_REG),
+            Ok(FunctionType::new_endo(vec![tv]))
+        );
+        Ok(())
+    }
 }
diff --git a/src/types/type_param.rs b/src/types/type_param.rs
index b87e2ad82..16d9e0a5a 100644
--- a/src/types/type_param.rs
+++ b/src/types/type_param.rs
@@ -97,6 +97,12 @@ impl From<TypeBound> for TypeParam {
     }
 }
 
+impl From<Type> for TypeArg {
+    fn from(ty: Type) -> Self {
+        Self::Type { ty }
+    }
+}
+
 /// A statically-known argument value to an operation.
 #[derive(Clone, Debug, PartialEq, Eq, serde::Deserialize, serde::Serialize)]
 #[non_exhaustive]