The Stacks project

Typo in (2): it should be “let $\mathcal{L}$ be an invertible $\mathcal{O}_Y$-module.”

Part (1) can be stated more generally as:

Let $f:X\to Y$ be a morphism of locally ringed spaces. Assume pullbacks of meromorphic functions are defined for $f$. Let $\mathcal{G}\in Mod(\mathcal{O}_X)$, $\mathcal{F}\in Mod(\mathcal{O}_Y)$ and suppose we have a morphism $\mathcal{F}\to f_*\mathcal{G}$ of $\mathcal{O}_Y$-modules. Then there is a canonical morphism $\mathcal{K}_Y(\mathcal{F})\to f_*\mathcal{K}_X(\mathcal{G})$ of $\mathcal{K}_Y$-modules. (In particular, setting $\mathcal{G}=f^*\mathcal{F}$, the map $\mathcal{F}\to f_*f^*\mathcal{F}$ to be the unit of $f_*\dashv f^*$ and taking global sections gives the pullback map pullback map $f^* : \Gamma(Y, \mathcal{K}_Y(\mathcal{F})) \to \Gamma(X, \mathcal{K}_X(f^*\mathcal{F}))$.)

The construction is the composite of canonical maps:

$$\mathcal{F}\otimes_{\mathcal{O}_Y}\mathcal{K}_Y \to f_*\mathcal{G}\otimes_{f_*\mathcal{O}_X}f_*\mathcal{K}_X \to f_*(\mathcal{G}\otimes_{\mathcal{O}_X}\mathcal{K}_X).$$

I regarded this more general form of part (1) useful for instance if one is working with the sheaf of meromorphic differentials on a $k$-variety $X$, i.e., $\mathcal{K}_X(\Omega_{X/k})$. Pullbacks of meromorphic functions are defined for the normalization $\nu:X'\to X$. Hence one obtains a morphism $\mathcal{K}_X(\Omega_{X/k})\to\nu_*\mathcal{K}_{X'}(\Omega_{X'/k})$. When $X$ is a curve, this last morphism is used by Serre in Algebraic Groups and Class Fields to define the “sheaf of regular meromorphic differentials.”

Also, in my reformulation of part (1), instead of "canonical" one can say that:

The map $\mathcal{K}_Y(\mathcal{F})\to f_*\mathcal{K}_X(\mathcal{G})$ is the unique morphism of $\mathcal{K}_Y$-modules that makes the following diagram commute: $$\require{AMScd} \begin{CD} \mathcal{F}@>>>f_*\mathcal{G}\\ @VVV@VVV\\ \mathcal{K}_Y(\mathcal{F})@>>> f_*\mathcal{K}_X(\mathcal{G}) \end{CD}$$

The fact that the constructed morphism indeed makes the square commute can be seen leveraging the factorization $\mathcal{F}\otimes_{\mathcal{O}_Y}\mathcal{K}_Y \to f_*\mathcal{G}\otimes_{f_*\mathcal{O}_X}f_*\mathcal{K}_X \to f_*(\mathcal{G}\otimes_{\mathcal{O}_X}\mathcal{K}_X)$.

Uniqueness follows from the fact that by precomposing the $\mathcal{K}_Y$-linear map $can:\mathcal{K}_Y(\mathcal{F})\to f_*\mathcal{K}_X(\mathcal{G})$ with the unit $\mathcal{F}\to\mathcal{K}_Y(\mathcal{F})$ we are obtaining the adjunct $\mathcal{O}_Y$-linear morphism $\mathcal{F}\to f_*\mathcal{K}_X(\mathcal{G})$ of $can$; thus, it is uniquely determined.

Regarding #8691: Ignore last paragraph. Serre does not use that map in his book. Also, the map $\mathcal{K}_X(\Omega_{X/k})\to\nu_*\mathcal{K}_{X'}(\Omega_{X'/k})$ from the case I explained is just the identity, as $\mathcal{K}_X(\Omega_{X/k})$ and $\mathcal{K}_{X'}(\Omega_{X'/k})$ are the sheaves constantly $\Omega_{X/k,\eta}\cong\Omega_{X'/k,\eta'}$, where $\eta\in X$, $\eta'\in X'$ is the generic point.