Let $U,V$ two open subset of a topological space $X$ with non-empty intersection. Let $F$ be a sheaf of commutative rings on $X$ and say $\rho$ are its transition maps
Let $f\in F(U\cup V)$. Is it true that $\rho^U_{U\cap V}(f)=\rho^V_{U\cap V}(f)$ ?? It should follow directly from the sheaves axioms but actually I can't see it.
Do you mean $$\rho_{U\cap V}^U(\rho_U^{U\cup V}(f)) =\rho_{U\cap V}^V(\rho_V^{U\cup V}(f))?$$ By the sheaf axioms, both sides are $\rho_{U\cap V}^{U\cup V}(f)$.