Let $C$ be a category, $I$ a small category and let $\alpha: I \rightarrow C$ be a diagram. Is it true that $$\lim: \operatorname{Fun}(I, C) \rightarrow C, \alpha \mapsto \lim(\alpha)$$ is a functor ? Can someone explain me why ?
Remark: $\operatorname{Fun}(I, C)$ denotes the category of functors from $I$ to $C$.
Thanks in advance.
First of all, for this to be well-defined, of course you need to assume that the necessary limits exist in $C$. So assume that this is the case in the following. (Edit: There are more set-theoretic subtleties, see the comments below. So I guess the best way to interpret this question without delving into that too much is to ask:
If $\lim(\alpha)$ and $\lim(\beta)$ exist for two diagrams and we chose a particular object for each (these are unique up to an isomorphism which is unique wrt to making the relevant diagrams commute) and have compatible maps between $\alpha$ and $\beta$ (i.e. natural transformations when we regard them as functors, why do we get an induced morphism $\lim(\alpha) \to \lim(\beta)$ and why do these induced morphisms behave nice with identities and composition? This is the question I shall consider below. Note that this way to state it is actually more general: we don't need the limits to exist for any possible diagram with the shape $I$, just for the diagrams we work with.)
Suppose that we have a morphism of functors $\alpha, \beta:I \to C$, i.e. a natural transformation $\eta: \alpha \Rightarrow \beta$.
By the universal property of a limit, we have a morphism $\lim(\alpha) \to \alpha(i)$ for each $i$ such that the necessary triangles commute, or in other words, we have a natural transformation from the constant functor $\underline{\lim(\alpha)}: I \to C$ which sends every object in $I$ to $\lim(\alpha)$ and every morphism to $id_{\lim(\alpha)}$ to $\alpha$. If we compose this natural transformation with $\eta$, we get a natural transformation $\underline{\lim(\alpha)} \Rightarrow \beta$. But a natural transformation from a constant functor to $\beta$ is the same as a cone over $\beta$, so by the universal property of the limit, we get a unique induced morphism $\lim{\alpha} \to \lim{\beta}$ such that the relevant diagrams commute, let's call this morphism $\lim{\eta}$
I'll leave it as an exercise to show that this behaves well with respect to identities and composition. The important part is to use the uniqueness of the induced morphism. If we have a sequence of natural transformations $\eta:\alpha \Rightarrow \beta$ and $\theta: \beta \Rightarrow \gamma$, then both $\lim{\theta} \circ \lim{\eta}$ and $\lim{(\theta \circ \eta)}$ are induced by the universal property of $\lim{\gamma}$ with respect to the same cone.