Limit as right adjoint functor

Let $I$$I$ be a small category and $\mathcal{C}$$\mathcal C$ be a complete category, and let us consider the functor category $\mathrm{Fun}(I,\mathcal{C})$$\mathrm{Fun}(I,\mathcal C)$.

Let us define $lim$$\lim$ as a functor from $\mathrm{Fun}(I,\mathcal{C})$$\mathrm{Fun}(I,\mathcal C)$ to $\mathcal{C}$$\mathcal C$ as follows:

For all $F\in \mathrm{Fun}(I,\mathcal{C})$$F\in\mathrm{Fun}(I,\mathcal C)$, we have $lim:F⟼limF\in \mathcal{C}$$\lim:F\longmapsto\lim F\in\mathcal C$. For a morphism $\eta :F\to G$$\eta:F\to G$, we induce the morphism $lim\eta :limF\to limG$$\lim\eta:\lim F\to\lim G$ via the universal property of $limG$$\lim G$. Since the natural transformation $\eta$$\eta$ will trans a cone over $F$$F$ to cone over $G$$G$. Easy to see that we do define a functor here.

Let us define its left adjoint functor $\mathrm{\Delta }:\mathcal{C}\to \mathrm{Fun}(I,\mathcal{C})$$\Delta:\mathcal C\to\mathrm{Fun}(I,\mathcal C)$as follows:

For all $X\in \mathcal{C}$$X\in\mathcal C$, we define $\mathrm{\Delta }(X)$$\Delta(X)$ to be the constant functor. i.e. $\mathrm{\forall }i\in \mathrm{Ob}(I),\mathrm{\Delta }(X)(i)=X$$\forall i\in \mathrm{Ob}(I),\Delta(X)(i)=X$ and maps all the morphism to ${\mathrm{id}}_{\mathrm{X}}$$\mathrm{id_X}$.

Proposition. $lim$$\lim$ is the right adjoint of $\mathrm{\Delta }$$\Delta$.

Proof.

We need to prove that

$${\mathrm{Hom}}_{\mathrm{Fun}(I,\mathcal{C})}(\mathrm{\Delta }(X),F)\cong {\mathrm{Hom}}_{\mathcal{C}}(X,limF)$$

Each $\eta :\mathrm{\Delta }(X)\to F$$\eta:\Delta(X)\to F$ gives you a cone over $F$$F$, which is bijectively and naturally correpsonds to a morphism from $X$$X$ to $limF.$$\lim F.$ $◻$$\square$