Let $d:\mathbf I^{\text{op}}\to \mathbf{Set}$ be a functor, with $\bf I$ a small category. Set $t:\mathbf I^{\text{op}}\to \mathbf{Set}$ the (constant) functor sending every object to the singleton. I read on nLab that the $\operatorname{lim}d$ can be defined as $\operatorname{Nat} (t,d)$. With this definition, the following should be a triviality ($S$ is any set): $$\operatorname{Hom}(S, \operatorname{lim} d)\cong \operatorname{lim}(\operatorname{Hom}(S,d(-)).$$
Is this the reason? $$ \operatorname{lim}(\operatorname{Hom}(S,d)\cong \operatorname{lim}\prod_{s\in S}d$$ $$\cong \operatorname{Nat}(t,\prod_{s\in S}d)\cong\prod_{s\in S}\operatorname{Nat}(t,d)$$ $$\cong \prod_{s\in S}\operatorname{lim}d\cong \operatorname{Hom}(S, \operatorname{lim} d).$$ To obtain it I mimicked the argument that I received in an answer recently, but in this context I have a doubt: for the third bijection I used that $\prod_{s\in S} d$, that is defined as $\prod_{s\in S} d(i)$ on every $i\in \operatorname {ob}( \mathbf I)$, has the universal property of $\prod_{s\in S}(d)_s$, the power of $d$ (as functor) over $S$. Indeed, I used that any natural transformation $t\to \prod_{s\in S}d$ is uniquely determined by the data of a natural transformation $t\to d$ for every $s\in S$. I.e., I used that ($*$) the limit (here the power) of a presheaf is computed objectwise, that however is a consequence of the bijection of the thesis. Anyway, to prove directly ($*$), in case of powers, should be straightforward, so my question is: do I need actually to prove that powers of presheaves are computed objectwise to prove the bijection of the thesis? Thanks for any clarify