We all know that the dual of the colimit of a diagram in the category of chain complexes (and similar categories) is the limit of the duals diagram. This follows immediately from the general fact that the $\hom$ functor sends colimits in the first slot to limits. I am confronted with a situation where I would like the opposite to be true, which sparked my interest about the most general context where this would happen.

Consider a diagram $D$ in the full subcategory category of chain complexes given by chain complexes "of finite type", by which I will mean chain complexes that have degrees bounded below (or: bounded above) and that are finite dimensional in every degree. In particular, these chain complexes have the property that they are isomorphic to their double duals through the canonical inclusion $v\mapsto\langle-, v\rangle$.

Suppose that this diagram $D$ has a limit $\lim D$ in the full subcategory of chain complexes of finite type. Examples would be finite products (trivial in what I want to do) or kernels (which interest me much more).

Then, the dual of the limit of the diagram is the colimit of the dual of the diagram. Indeed, we can consider the colimit $\operatorname{colim}D^\vee$ of the dual diagram $D^\vee$ and take its dual, which gives us the limit of the double dual diagram. But $$\left(\operatorname{colim}D^\vee\right)^\vee\cong\lim D^{\vee\vee}\cong\lim D$$ since $D$ is in the subcategory of finite type. Notice that his in particular implies that $\operatorname{colim}D^\vee$ is of finite type (as taking duals can only increase dimensions). Then $$\left(\lim D\right)^\vee\cong\left(\operatorname{colim}D^\vee\right)^{\vee\vee}\cong \operatorname{colim}D^\vee.$$ Is there a nice and general category theoretical explanation for this phenomenon? What are (reasonably) general situations where something like this occurs, and could someone provide a reference?

Also more generally: when are duals of limits the colimit of the duals?

We all know that the dual of the colimit of a diagram in the category of chain complexes (and similar categories) is the limit of the duals diagram. This follows immediately from the general fact that the $\hom$ functor sends colimits in the first slot to limits. I am confronted with a situation where I would like the opposite to be true, which sparked my interest about the most general context where this would happen.

Consider a diagram $D$ in the full subcategory category of chain complexes given by chain complexes "of finite type", by which I will mean chain complexes that have degrees bounded below (or: bounded above) and that are finite dimensional in every degree. In particular, these chain complexes have the property that they are isomorphic to their double duals through the canonical inclusion $v\mapsto\langle-, v\rangle$.

Suppose that this diagram $D$ has a limit $\lim D$ in the category of chain complexes and that this limit is in the full subcategory of chain complexes of finite type. Examples would be finite products (trivial in what I want to do) or kernels (which interest me much more).

Then, the dual of the limit of the diagram is the colimit of the dual of the diagram. Indeed, we can consider the colimit $\operatorname{colim}D^\vee$ of the dual diagram $D^\vee$ and take its dual, which gives us the limit of the double dual diagram. But $$\left(\operatorname{colim}D^\vee\right)^\vee\cong\lim D^{\vee\vee}\cong\lim D$$ since $D$ is in the subcategory of finite type. Notice that his in particular implies that $\operatorname{colim}D^\vee$ is of finite type (as taking duals can only increase dimensions). Then $$\left(\lim D\right)^\vee\cong\left(\operatorname{colim}D^\vee\right)^{\vee\vee}\cong \operatorname{colim}D^\vee.$$ Is there a nice and general category theoretical explanation for this phenomenon? What are (reasonably) general situations where something like this occurs, and could someone provide a reference?

Also more generally: when are duals of limits the colimit of the duals?

We all know that the dual of the colimit of a diagram in the category of chain complexes (and similar categories) is the limit of the duals diagram. I am confronted with a situation where I would like the opposite to be true, which sparked my interest about the most general context where this would happen.

Consider a diagram $D$ in the full subcategory category of chain complexes given by chain complexes "of finite type", by which I will mean chain complexes that have degrees bounded below (or: bounded above) and that are finite dimensional in every degree. In particular, these chain complexes have the property that they are isomorphic to their double duals through the canonical inclusion $v\mapsto\langle-, v\rangle$.

Suppose that this diagram $D$ has a limit $\lim D$ in the full subcategory of chain complexes of finite type. Examples would be finite products (trivial in what I want to do) or kernels (which interest me much more).

Then, the dual of the limit of the diagram is the colimit of the dual of the diagram. Indeed, we can consider the colimit $\operatorname{colim}D^\vee$ of the dual diagram $D^\vee$ and take its dual, which gives us the limit of the double dual diagram. But $$\left(\operatorname{colim}D^\vee\right)^\vee\cong\lim D^{\vee\vee}\cong\lim D$$ since $D$ is in the subcategory of finite type. Notice that his in particular implies that $\operatorname{colim}D^\vee$ is of finite type (as taking duals can only increase dimensions). Then $$\left(\lim D\right)^\vee\cong\left(\operatorname{colim}D^\vee\right)^{\vee\vee}\cong \operatorname{colim}D^\vee.$$ Is there a nice and general category theoretical explanation for this phenomenon? What are (reasonably) general situations where something like this occurs, and could someone provide a reference?

Also more generally: when are duals of limits the colimit of the duals?

We all know that the dual of the colimit of a diagram in the category of chain complexes (and similar categories) is the limit of the duals diagram. This follows immediately from the general fact that the $\hom$ functor sends colimits in the first slot to limits. I am confronted with a situation where I would like the opposite to be true, which sparked my interest about the most general context where this would happen.

Consider a diagram $D$ in the full subcategory category of chain complexes given by chain complexes "of finite type", by which I will mean chain complexes that have degrees bounded below (or: bounded above) and that are finite dimensional in every degree. In particular, these chain complexes have the property that they are isomorphic to their double duals through the canonical inclusion $v\mapsto\langle-, v\rangle$.

Suppose that this diagram $D$ has a limit $\lim D$ in the full subcategory of chain complexes of finite type. Examples would be finite products (trivial in what I want to do) or kernels (which interest me much more).

Then, the dual of the limit of the diagram is the colimit of the dual of the diagram. Indeed, we can consider the colimit $\operatorname{colim}D^\vee$ of the dual diagram $D^\vee$ and take its dual, which gives us the limit of the double dual diagram. But $$\left(\operatorname{colim}D^\vee\right)^\vee\cong\lim D^{\vee\vee}\cong\lim D$$ since $D$ is in the subcategory of finite type. Notice that his in particular implies that $\operatorname{colim}D^\vee$ is of finite type (as taking duals can only increase dimensions). Then $$\left(\lim D\right)^\vee\cong\left(\operatorname{colim}D^\vee\right)^{\vee\vee}\cong \operatorname{colim}D^\vee.$$ Is there a nice and general category theoretical explanation for this phenomenon? What are (reasonably) general situations where something like this occurs, and could someone provide a reference?

Also more generally: when are duals of limits the colimit of the duals?