Let $f\colon X\to Y$ be a continuous map of 'nice' topological spaces (e.g. $f$ is a smooth map of smooth manifolds; $f$ might be assumed to be proper although I am not sure it is relevant). Let $\underline{\mathbb{F}}_X$ be the constant sheaf on $X$ with coefficients in a field $\mathbb{F}$.

Does the obvious morphism of sheaves $\underline{\mathbb{F}}_X\otimes \underline{\mathbb{F}}_X\to \underline{\mathbb{F}}_X$ induce a canonical morphism in the derived category $D^+(Sh(\mathbb{F}))$ of sheaves of $\mathbb{F}$-vector spaces $$Rf_*(\underline{\mathbb{F}}_X)\otimes Rf_*(\underline{\mathbb{F}}_X)\to Rf_*(\underline{\mathbb{F}}_X),$$ where $Rf_*$ is the push-forward functor. If yes, is this morphism associative?

Remark. When $Y$ is a point the required map does exist and is just the multiplication map in cohomology $H^*(X,\mathbb{F})$.

Let $f\colon X\to Y$ be a continuous map of 'nice' topological spaces (e.g. $f$ is a smooth map of smooth manifolds; $f$ might be assumed to be proper although I am not sure it is relevant). Let $\underline{\mathbb{F}}_X$ be the constant sheaf on $X$ with coefficients in a field $\mathbb{F}$.

Does the obvious morphism of sheaves $\underline{\mathbb{F}}_X\otimes \underline{\mathbb{F}}_X\to \underline{\mathbb{F}}_X$ induce a canonical morphism in the derived category $D^+(Sh_\mathbb{F})$ of sheaves of $\mathbb{F}$-vector spaces $$Rf_*(\underline{\mathbb{F}}_X)\otimes Rf_*(\underline{\mathbb{F}}_X)\to Rf_*(\underline{\mathbb{F}}_X),$$ where $Rf_*$ is the push-forward functor. If yes, is this morphism associative?

Remark. When $Y$ is a point the required map does exist and is just the multiplication map in cohomology $H^*(X,\mathbb{F})$.

Let $f\colon X\to Y$ be a continuous map of 'nice' topological spaces (e.g. $f$ is a smooth map of smooth manifolds; $f$ might be assumed to be proper although I am not sure it is relevant). Let $\underline{\mathbb{F}}_X$ be the constant sheaf on $X$ with coefficients in a field $\mathbb{F}$.

Does the obvious morphism of sheaves $\underline{\mathbb{F}}_X\otimes \underline{\mathbb{F}}_X\to \underline{\mathbb{F}}_X$ induce a canonical morphism in the derived category $D^+(Sh(\mathbb{F}))$ of sheaves of $\mathbb{F}$-vector spaces $$Rf_*(\underline{\mathbb{F}}_X)\otimes Rf_*(\underline{\mathbb{F}}_X)\to Rf_*(\underline{\mathbb{F}}_X),$$ where $Rf_*$ is the push-forward functor. If yes, is this morphism associative?

Let $f\colon X\to Y$ be a continuous map of 'nice' topological spaces (e.g. $f$ is a smooth map of smooth manifolds; $f$ might be assumed to be proper although I am not sure it is relevant). Let $\underline{\mathbb{F}}_X$ be the constant sheaf on $X$ with coefficients in a field $\mathbb{F}$.

Does the obvious morphism of sheaves $\underline{\mathbb{F}}_X\otimes \underline{\mathbb{F}}_X\to \underline{\mathbb{F}}_X$ induce a canonical morphism in the derived category $D^+(Sh(\mathbb{F}))$ of sheaves of $\mathbb{F}$-vector spaces $$Rf_*(\underline{\mathbb{F}}_X)\otimes Rf_*(\underline{\mathbb{F}}_X)\to Rf_*(\underline{\mathbb{F}}_X),$$ where $Rf_*$ is the push-forward functor. If yes, is this morphism associative?

Remark. When $Y$ is a point the required map does exist and is just the multiplication map in cohomology $H^*(X,\mathbb{F})$.

Let $f\colon X\to Y$ be a continuous map of 'nice' topological spaces (e.g. $f$ is a smooth map of smooth manifolds; $f$ might be assumed to be proper although I am not sure it is relevant). Let $\underline{\mathbb{F}}_X$ be the constant sheaf on $X$ with coefficients in a field $\mathbb{F}$.

Does the obvious map $\underline{\mathbb{F}}_X\otimes \underline{\mathbb{F}}_X\to \underline{\mathbb{F}}_X$ induce a canonical morphism $$Rf_*(\underline{\mathbb{F}}_X)\otimes Rf_*(\underline{\mathbb{F}}_X)\to Rf_*(\underline{\mathbb{F}}_X),$$ where $Rf_*$ is the push-forward functor in the derived category $D^+(Sh(\mathbb{F}))$ of sheaves of $\mathbb{F}$-vector spaces. If yes, is this morphism associative?

Let $f\colon X\to Y$ be a continuous map of 'nice' topological spaces (e.g. $f$ is a smooth map of smooth manifolds; $f$ might be assumed to be proper although I am not sure it is relevant). Let $\underline{\mathbb{F}}_X$ be the constant sheaf on $X$ with coefficients in a field $\mathbb{F}$.

Does the obvious morphism of sheaves $\underline{\mathbb{F}}_X\otimes \underline{\mathbb{F}}_X\to \underline{\mathbb{F}}_X$ induce a canonical morphism in the derived category $D^+(Sh(\mathbb{F}))$ of sheaves of $\mathbb{F}$-vector spaces $$Rf_*(\underline{\mathbb{F}}_X)\otimes Rf_*(\underline{\mathbb{F}}_X)\to Rf_*(\underline{\mathbb{F}}_X),$$ where $Rf_*$ is the push-forward functor. If yes, is this morphism associative?