Global Section of $\mathcal{F}$ induces a Global Section of $f^*\mathcal{F}$

Question

Let $f: X \to Y$ a morphism of schemes, $\mathcal{F}$ a sheaf on $Y$ and $s \in \Gamma(Y, \mathcal{F})$ a global section.

My question is how does $s$ induce a global section in $\Gamma(X, f^*\mathcal{F})$ ?

Answer 1

You have maps $$\Gamma(Y,\mathcal{F}) \to \Gamma(Y,f_*f^*\mathcal{F}) = \Gamma(X,f^*\mathcal{F}),$$ where the first map is induced by the unit of adjunction $\eta_\mathcal{F}: \mathcal{F} \to f_*f^*\mathcal{F}.$

Answer 2

By definition, for any open set $U$ of $X$, $f^* \mathcal F(U)$ is the direct limit of $\mathcal (V)$ where $V$ contains $f(U)$. Thus any $s_V \in \mathcal F(V)$ for $V \supset f(U)$ defines an element $[s_V] \in f^* \mathcal F(U)$.