Colimit functor on an enriched category

Question

Let $\mathscr{M}$ be a cocomplete category enriched over topological spaces, and $J$ be a small (ordinary) category toplogized with the discrete topology. Is it true that the functor $Fun(J,\mathscr{M})\rightarrow \mathscr{M}$ taking the colimit of the diagram is a functor of categories enriched over topological spaces?