Remark 25.12.7. Let $\mathcal{C}$ be a site. Let $K$ and $L$ be objects of $\text{SR}(\mathcal{C})$. Write $K = \{ U_ i\} _{i \in I}$ and $L = \{ V_ j\} _{j \in J}$. Assume $U = \coprod _{i \in I} U_ i$ and $V = \coprod _{j \in J} V_ j$ exist. Then we get

$\mathop{\mathrm{Mor}}\nolimits _{\text{SR}(\mathcal{C})}(K, L) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U, V)$

as follows. Given $f : K \to L$ given by $\alpha : I \to J$ and $f_ i : U_ i \to V_{\alpha (i)}$ we obtain a transformation of functors

$\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, -) = \prod \nolimits _{j \in J} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V_ j, -) \to \prod \nolimits _{i \in I} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U_ i, -) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U, -)$

sending $(g_ j)_{j \in J}$ to $(g_{\alpha (i)} \circ f_ i)_{i \in I}$. Hence the Yoneda lemma produces the corresponding map $U \to V$. Of course, $U \to V$ maps the summand $U_ i$ into the summand $V_{\alpha (i)}$ via the morphism $f_ i$.

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