large cocomplete category and thin category

Let \(\mathcal{C}\) be a category, such that \(\mathcal{C}\) admits \(\vert \mathrm{Mor}(\mathcal{C}) \vert\) sized coproducts.
Then \(\mathcal{C}\) is a thin category

Let \(f,g: c_1 \rightarrow c_2\) be different morphisms.
Let \(J \coloneqq \mathrm{Mor}(\mathcal{C})\)
Then consider

\begin{align*} \coprod_{J} c_1 \end{align*}

Then there exists a bijection

\begin{align*} \mathrm{Hom}_{\mathrm{Fun}(J^{\delta}, \mathcal{C})}(\mathrm{const}(c_1), \mathrm{const}(c_2)) \cong \mathrm{Hom}_{\mathcal{C}}(\coprod_{J} c_1,c_2) \hookrightarrow \mathrm{Mor}(\mathcal{C}) \end{align*}

The left has at least cardinality \(\mathcal{P}(J)\), as for each \(j \in J\) we may choose one of \(f\) or \(g\).
But the right side has cardinality of at most \(J\)

contradiction to Satz von Cantor