# How to arrive at unique factorization through the limit given naturality compatibility conditions?

If $$\alpha : I \to C$$ from a small category to any category $$C$$. Define a functor $$\lim\limits_{\rightarrow} \alpha : X \mapsto \lim\limits_{\leftarrow} \text{Hom}_C(\alpha, X)$$ from $$C^{op}$$ to $$\text{Set}$$.

If $$\lim\limits_{\rightarrow} \alpha$$ is representable then let $$Y$$ be a representative object. Then we have $$\text{Hom}_{C}(Y, Y) \simeq \lim\limits_{\leftarrow} \text{Hom}_C(\alpha, Y)$$ by definition of representable. So that to $$\text{id}_Y$$ is a associated a natural map $$\rho$$ in $$\lim\limits_{\leftarrow} \text{Hom}_C(\alpha, Y)$$ such that $$\rho_j \circ \alpha(s) = \rho_i$$ for any $$s : i \to j$$ in $$I$$.

Suppose that we are given another family of morphism $$f_i : \alpha(i) \to X$$ in $$C$$ such that $$f_j \circ \alpha(s) = f_i$$. I'm seeing how there exists a unique map $$g$$ in $$\text{Hom}_C(Y, X)$$ but I'm not seeing how $$f_i = g\circ \rho_i$$.

• The functor $\varprojlim\text{Hom}_C(\alpha,X)$ is from $C$ to $\mathbf{Set}$, not from $C^{op}$. – Oskar Nov 24 '18 at 13:03