Let $I$ and $C$ be categories, assume $I$ is small and denote $\Delta$ the functor $C \to C^I$ that sends $Y\in C$ to the constant functor $I\to C$, i.e. $\Delta(Y)(i) = Y$ and $(i \to j) \mapsto \mathrm{id}_Y$ for $i, j \in I$. If we assume that any functor $I\to C$ admits a colimit, I’d like to show that, for $\alpha$ a functor $I\to C$ and $Y$ an object in $C$: $$\newcommand{\Hom}{\operatorname{Hom}} \newcommand{\colim}{\operatorname{colim}} \Hom_C(\colim \alpha, Y) \simeq \Hom_{\mathrm{Fct}(I, C)}(\alpha, \Delta Y) \,.$$
I have the isomorphism $\Hom_C(\colim \alpha, Y) \simeq \lim \Hom_C(\alpha, Y)$, but after that I’m not sure how to continue. My intuition tells me there’s probably a string of natural isomorphisms that would lead to the desired results but I’m not sure. This is new material to me I’m learning on my own so it gets a bit confusing every now and then.
EDIT: I’ve realised this would amount to showing that $\colim$ and $\Delta$ are a pair of adjoint functors but this doesn’t necessarily help me any further.
This is essentially definitional, but the "essentially" carries a lot of weight when you're new to the concepts. Probably the key step you need is to write down $\mathrm{lim}_I\mathrm{Hom}_C(\alpha(i),Y)$ explicitly as a certain subset of the product of those homsets. Similarly, a natural transformation $\alpha\to \Delta Y$ is a certain subset of the product of the homsets $\mathrm{Hom}_C(\alpha(i),(\Delta Y)(i))$...I also added a couple more details to the decoration where I thought your notational abuses were risking confusion.