Limit preserving inclusion of a subcategory

Question

Is every inclusion functor of a full subcategory limit preserving?

Answer 1

No. Nor is it colimit preserving. What is true is that the inclusion reflects all limits and colimits. That is, if $\mathcal{D}$ is a full subcategory of $\mathcal{C}$ with inclusion $i : \mathcal{D} \hookrightarrow \mathcal{C}$, and $A : \mathcal{J} \to \mathcal{D}$ is a diagram, if there is an object $L$ in $\mathcal{D}$ such that $i L = \varprojlim i A$, then $L = \varprojlim A$.