Suppose $k$ is a field which admits an embedding $\sigma:k \hookrightarrow \mathbb{C}$. If $X$ is a smooth projective varieties over $k$, from Hodge theory there is a natural pure Hodge structure on \begin{equation} H^*(X(\mathbb{C}),\mathbb{Q})=\oplus_{i=0}^{2\text{dim}\,X}H^i(X(\mathbb{C}),\mathbb{Q}) \end{equation} and we will also denote this pure Hodge structure by $H^*(X(\mathbb{C}),\mathbb{Q})$. Suppose $Y$ is another smooth projective variety defined over $k$, the following post
seems to imply that any morphism (in the category of pure Hodge structures) from $H^*(X(\mathbb{C}),\mathbb{Q})$ to $H^*(Y(\mathbb{C}),\mathbb{Q})$ is induced by a Hodge class of $H^{2 \text{dim}\,X}((X \times_k Y)(\mathbb{C}),\mathbb{Q})$, i.e. an element of $H^{2 \text{dim}\,X}((X \times_k Y)(\mathbb{C}),\mathbb{Q}) \cap H^{ \text{dim}\,X,\text{dim}\,X}((X \times_k Y)(\mathbb{C}))$. E.g, a map $f: Y \rightarrow X$ determines a cycle $C^{\text{dim}\,X}(X \times_k Y)$, which induces a Hodge cycle that will give such a morphism.
Any one who can explain this claim?