Direct image preserves sheaves and objectwise algebraic structure

Let $\mathcal{F}$ be a sheaf on $X$, let $V={\bigcup }_i{V}_i$ be an open cover in $Y$, and let $s_i\in (f_{\ast }\mathcal{F})(V_i)=\mathcal{F}(f^{-1}(V_i))$ be compatible on overlaps. By [F1], the sets $f^{-1}(V_i)$ cover $f^{-1}(V)$, so [L1] gives a unique section $s\in \mathcal{F}(f^{-1}(V))$ restricting to every $s_i$. This section is exactly an element of $(f_{\ast }\mathcal{F})(V)$, so $f_{\ast }\mathcal{F}$ is a sheaf.

If $\mathcal{F}$ is a sheaf of groups, rings, or modules, then each section set of $f_{\ast }\mathcal{F}$ is literally the corresponding section set of $\mathcal{F}$ over a preimage open set. Hence the algebraic operations are inherited objectwise, and the restriction maps are the same homomorphisms as before. By [L2] and step 1.1, $f_{\ast }\mathcal{F}$ is a sheaf of the same kind.

Steps 1.1 and 2.1 prove both assertions.