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 $f_i = g\circ \rho_i$.

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

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