In the category of sets and set-theoretic functions, every arrow can be obtained by only using the universal property of the small coproducts of $1=\{ 0 \}$ applied to the coprojections that exhibit them as a coproducts. In the following sense.
Let $f \colon X \to Y$ be a map between sets. Then $X$ is the coproduct of a $X$-indexed family of $1$'s, that is, $X=\sum_{x \in X}1$, and the same holds for $Y=\sum_{y \in Y}1$. Let $\{\iota_x\colon 1 \to X\}_{x \in X}$ and $\{\eta_y\colon 1\to Y \}_{y \in Y}$ be the coprojections that exhibit $X$ and $Y$ as those coproducts (respectively). Since $\{\eta_{f(x)} \}_{x \in X}$ is a cocone of the family $\{1_x=1 \}_{x \in X}$, we get unique a map $g \colon X \to Y$ such that $g \circ \iota_x =\eta_{f(x)}$, for every $x \in X$. The point is that, clearly, this map $g$ happens to be $f$. Hence, we see that every arrow $X \to Y$ can be obtained by invoking the universal property of the coproduct $X=\sum_{x\in X}1$ on an $X$-indexed subfamily of $\{\eta_y\colon 1\to Y \}_{y \in Y}$ (noticing that it is as cocone of $\{1_x=1 \}_{x \in X}$).
I was wondering if this also holds for an arbitrary topos $\mathcal{E}$. In detail, if $X$ and $Y$ are sets and $1$ is the terminal object of $\mathcal{E}$, is it the case that every arrow $\sum_{x \in X}1\to\sum_{y \in Y}1$ [assuming that these coproducts exist or that, for instance, $\mathcal{E}$ is cocomplete] can be obtained by invoking the universal property of the coproduct $\sum_{x \in X}1$ on an $X$-indexed subfamily of the family of coprojections that exhibit $\sum_{y\in Y}1$ as a coproduct (suppose that we fixed such a family before for both $\sum_{x \in X}1$ and $\sum_{y \in Y}1$)?
If $\phi$ is an arrow $\sum_{x \in X}1\to\sum_{y \in Y}1$ and $\{\iota_x\colon 1 \to \sum_{x \in X}\}_{x \in X}$ and $\{\eta_y\colon 1\to \sum_{y \in Y}1 \}_{y \in Y}$ are our fixed families of coprojections, we can consider the arrows $\phi \circ \iota_x \colon 1 \to \sum_{1 \in Y}1$, for $x \in X$. Clearly, the answer would be yes if we proved that, for every $x\in X$, there is $y \in Y$ such that $\phi \circ \iota_x=\eta_y$. Is this the case?