diff --git a/src/lax/category.rs b/src/lax/category.rs
index f75d4b3..956561c 100644
--- a/src/lax/category.rs
+++ b/src/lax/category.rs
@@ -32,6 +32,18 @@ impl<O: Clone + PartialEq, A: Clone> Arrow for OpenHypergraph<O, A> {
         if self.target() != other.source() {
             return None;
         }
+        self.lax_compose(other)
+    }
+}
+
+impl<O: Clone + PartialEq, A: Clone> OpenHypergraph<O, A> {
+    /// Compose two open hypergraphs, unifying the boundary nodes without checking labels.
+    ///
+    /// Returns None when the boundary arities do not match.
+    pub fn lax_compose(&self, other: &Self) -> Option<Self> {
+        if self.targets.len() != other.sources.len() {
+            return None;
+        }
 
         let n = self.hypergraph.nodes.len();
         let mut f = self.tensor(other);
@@ -39,7 +51,7 @@ impl<O: Clone + PartialEq, A: Clone> Arrow for OpenHypergraph<O, A> {
             f.unify(*u, NodeId(v.0 + n));
         }
 
-        f.sources = f.sources[..self.sources.len()].to_vec();
+        f.sources.truncate(self.sources.len());
         f.targets = f.targets[self.targets.len()..].to_vec();
 
         Some(f)
diff --git a/tests/lax/mod.rs b/tests/lax/mod.rs
index a3a11c8..2e1d7c6 100644
--- a/tests/lax/mod.rs
+++ b/tests/lax/mod.rs
@@ -1,2 +1,3 @@
 pub mod eval;
 pub mod hypergraph;
+pub mod open_hypergraph;
diff --git a/tests/lax/open_hypergraph.rs b/tests/lax/open_hypergraph.rs
new file mode 100644
index 0000000..188f033
--- /dev/null
+++ b/tests/lax/open_hypergraph.rs
@@ -0,0 +1,47 @@
+use open_hypergraphs::category::Arrow;
+use open_hypergraphs::lax::{NodeId, OpenHypergraph};
+
+#[derive(Clone, Debug, PartialEq)]
+enum Obj {
+    A,
+    B,
+    C,
+}
+
+#[derive(Clone, Debug, PartialEq)]
+enum Op {
+    F,
+    G,
+}
+
+#[test]
+fn test_compose_type_mismatch() {
+    let f = OpenHypergraph::singleton(Op::F, vec![Obj::A], vec![Obj::B]);
+    let g = OpenHypergraph::singleton(Op::G, vec![Obj::C], vec![Obj::A]);
+
+    assert!(Arrow::compose(&f, &g).is_none());
+}
+
+#[test]
+fn test_lax_compose_allows_mismatch_and_unifies_boundary() {
+    let f = OpenHypergraph::singleton(Op::F, vec![Obj::A], vec![Obj::B]);
+    let g = OpenHypergraph::singleton(Op::G, vec![Obj::C], vec![Obj::A]);
+
+    let composed = f.lax_compose(&g).expect("arity matches");
+
+    assert_eq!(composed.sources, vec![NodeId(0)]);
+    assert_eq!(composed.targets, vec![NodeId(3)]);
+    assert_eq!(composed.hypergraph.quotient.0, vec![NodeId(1)]);
+    assert_eq!(composed.hypergraph.quotient.1, vec![NodeId(2)]);
+}
+
+#[test]
+fn test_compose_matches_lax_compose_on_equal_boundaries() {
+    let f = OpenHypergraph::singleton(Op::F, vec![Obj::A], vec![Obj::B]);
+    let g = OpenHypergraph::singleton(Op::G, vec![Obj::B], vec![Obj::C]);
+
+    let strict = Arrow::compose(&f, &g).expect("labels match");
+    let lax = f.lax_compose(&g).expect("arity matches");
+
+    assert_eq!(strict, lax);
+}