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 finite 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:
[Colimits in the category of smooth manifolds](https://mathoverflow.net/questions/19116/colimits-in-the-category-of-smooth-manifolds)


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


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



inclusions of linear colimits into smooth manifolds
The canonical map $i(\mathbb{R}^n) \coprod\_M i(\mathbb{R}^m) \to i(\mathbb{R}^n \coprod\_V \mathbb{R}^m)$, where the coproduct index indicates the ambient category, corresponds to the smooth map $\mathbb{R}^n \sqcup \mathbb{R}^m \to \mathbb{R}^{n+m}$. It is neither surjective nor injective (the two zero vectors are mapped to the zero vector). So $i$ doesn't preserve coproducts.


The problem is already that $i$ maps the initial vector space to the point, which is the terminal manifold, but not the initial manifold ($\emptyset$).


