Let $V$ be the category of finite dimensional vector spaces and $M$ the category of
smooth finite dimensional Hausdorff manifolds.

Now suppose any finite dimensional vector space is equipped with a smooth structure in such a way that any $n$-dimensional vector space is diffeomorph to $\mathbb{R}^n$
seen as a smooth manifold with the standard smooth structure.

This way there is a faithfull inclusion $\imath: V \to M$ by just forgetting the linear
structure. 

Now recall that $V$ is cocomplete while $M$ is not.  

To see that colimits exist in $V$ let $D : I \to V$ be a diagram with a small index
category $I$. To construct the colimit, let $h_i : D_i \to \bigoplus_{j \in I} D_j$
be the inclusions and $Q$
be the submodule generated by the images of the maps $h_i \circ Dd - h_j$ for
each morphism $d : j \to i$, and let $C = \bigoplus_{j\in I} D_j /Q$ be the quotient space. Then $(D_i \overset{q\circ h_i}{\to} C)_{i \in I}$ is a colimit of $D$, where $q$ is the quotient map.

Counterexamples to the existence of all colimits in $M$ are given here on MO for example at:
http://mathoverflow.net/questions/19116/colimits-in-the-category-of-smooth-manifolds


Now the question is: Does the inclusion $i: V \to M$ preserves colimits?


Obviously $(D_i \overset{q\circ h_i}{\to} C)_{i \in I}$ is a cocone in $M$, but is it sill universal?