diff --git a/M2/Macaulay2/m2/matrix.m2 b/M2/Macaulay2/m2/matrix.m2
index 66ec39c5a3..fbe2059657 100644
--- a/M2/Macaulay2/m2/matrix.m2
+++ b/M2/Macaulay2/m2/matrix.m2
@@ -625,11 +625,28 @@ isSubquotient(Module,Module) := (M,N) -> (
      relations N % relations M == 0
      )
 
+-----------------------------------------------------------------------------
+-- inducedMap
+-----------------------------------------------------------------------------
+
 inducedMap = method (
      Options => {
 	  Verify => true,
 	  Degree => null 
 	  })
+-- TODO: hookify this, so people can add more application specific induced maps
+inducedMap(Module, Module)          := Matrix => opts -> (M, N) -> (
+    if ambient M =!= ambient N then error "inducedMap: expected modules with same ambient free module";
+    -- e.g. avoid a gb computation for inducedMap(M, image basis(d, M))
+    if N.cache.?Monomials and M === target N.cache.Monomials
+    then map(M, N, N.cache.Monomials, Degree => opts.Degree)
+    else inducedMap(M, N, id_(ambient N), opts))
+inducedMap(Module, Nothing, Matrix) := Matrix => opts -> (M, N, f) -> (
+    B := image f;
+    -- e.g. avoid a gb computation for inducedMap(image f, , f)
+    if M === target B.cache.Monomials
+    then map(M, source B.cache.Monomials, B.cache.Monomials, Degree => opts.Degree)
+    else inducedMap(M, source f, f, opts))
 inducedMap(Module,Module,Matrix) := Matrix => opts -> (N',M',f) -> (
      N := target f;
      M := source f;
@@ -651,7 +668,6 @@ inducedMap(Module,Module,Matrix) := Matrix => opts -> (N',M',f) -> (
 	  if not isWellDefined f' then error "inducedMap: expected matrix to induce a well-defined map";
 	  );
      f')
-inducedMap(Module,Nothing,Matrix) := o -> (M,N,f) -> inducedMap(M,source f, f,o)
 inducedMap(Nothing,Module,Matrix) := o -> (M,N,f) -> inducedMap(target f,N, f,o)
 inducedMap(Nothing,Nothing,Matrix) := o -> (M,N,f) -> inducedMap(target f,source f, f,o)
 
@@ -665,10 +681,6 @@ addHook((inducedMap, Module, Module, Matrix), Strategy => Default, (opts, N', M'
      f' = map(N',M',f',Degree => if opts.Degree === null then degree f else opts.Degree);
      (f', g, gbN', gbM)))
 
-inducedMap(Module,Module) := Matrix => o -> (M,N) -> (
-    if ambient M =!= ambient N then error "inducedMap: expected modules with same ambient free module";
-     inducedMap(M,N,id_(ambient N),o))
-
 -- TODO: deprecate this in favor of isWellDefined
 inducesWellDefinedMap = method(TypicalValue => Boolean)
 inducesWellDefinedMap(Module,Module,Matrix) := (M,N,f) -> (
@@ -685,6 +697,8 @@ inducesWellDefinedMap(Module,Nothing,Matrix) := (M,N,f) -> inducesWellDefinedMap
 inducesWellDefinedMap(Nothing,Module,Matrix) := (M,N,f) -> inducesWellDefinedMap(target f,N,f)
 inducesWellDefinedMap(Nothing,Nothing,Matrix) := (M,N,f) -> true
 
+-----------------------------------------------------------------------------
+
 vars Ring := Matrix => R -> (
      g := generators R;
      if R.?vars then R.vars else R.vars =