Surjectiveness of convolution

Question

Consider the convolution mapping $j^*: Hom(X, Y) \otimes X \to Y$, given by bilinear formula $(\phi, x) \mapsto \phi(x)$, in a category of coherent sheaves or, generally, in any abelian category. I believe that it is surjective, can you help me to prove it?

Answer 1

This is not sujective, for example in $\mathbf{Ab}$, $$Hom(\Bbb{Z}/n,\Bbb{Z}) \otimes \Bbb{Z}/n \to \Bbb{Z}$$ has trivial image.

(You could even take $X=0$ for a trivial counter-example in any abelian category).