Construct natural transformation $u^* R^i f_* \rightarrow R^i g_* v^*$ without assumption of quasi-coherence I am reading Hartshorne Algebraic geometry. Chapter 3 Proposition 9.3 (in particular remark 9.3.1). It states that if we have commutative diagram in category of schemes (namely morphisms $f, g, h, u$ such that $ug = fv$ ) then we have natural transformation $u^* R^i f_* \rightarrow R^i g_* v^*$. Hartshorne argument (for existence of such natural transformation) need for all sheaves to be quasi-coherent. This seems strange for me cause this question make sense for general sheaves.
Can you construct this natural transformation without assumption of quasi-coherence?
 A: Yes, you can construct such a comparison transformation in any context of adjoint functors:  Suppose $$(\ast)\quad\quad\quad\begin{array}{ccc} {\mathscr C} & \xrightarrow{\alpha_{\ast}} & {\mathscr D} \\ {\scriptsize\beta_{\ast}}\downarrow & &\downarrow{\scriptsize\gamma_{\ast}} \\\ {\mathscr E} & \xrightarrow{\delta_{\ast}} & {\mathscr F}\end{array}$$ is a diagram of right adjoint functors equipped with a natural transformation $T: \gamma_{\ast}\alpha_{\ast}\Rightarrow \delta_{\ast}\beta_{\ast}$ (for example, $T$ might witness the commutativity up to isomorphism of the diagram). 
Then, denoting their left adjoints by $(-)^{\ast}$, there is a natural transformation $$\delta^{\ast}\gamma_{\ast}\Rightarrow \beta_{\ast} \alpha^{\ast},$$
given by the composition $$\delta^{\ast}\gamma_{\ast}\stackrel{\text{unit}}{\Rightarrow}\delta^{\ast}\gamma_{\ast}\alpha_{\ast}\alpha^{\ast}\Rightarrow\delta^{\ast}\delta_{\ast}\beta_{\ast}\alpha^{\ast}\stackrel{\text{counit}}{\Rightarrow}\beta_{\ast}\alpha^{\ast}.$$
This would apply for example to commutative squares $$\begin{array}{ccc} X & \stackrel{f}{\to} & Y \\ {\scriptsize g}\downarrow && \downarrow{\scriptsize h}\\ Z &\stackrel{w}{\to} & W\end{array}$$ of ringed topological spaces inducing a square $(\ast)$ when replacing each space by its category of sheaves of modules or the derived category of the latter. In the derived setting, note that for non-flat morphisms you have to derive the pullback functors as well. 
If you don't want to talk about derived categories and stick to your situation, you might also first construct $u^{\ast} f_{\ast}\Rightarrow g_{\ast} v^{\ast}$ by hand, from the underived adjunction $(-)^{\ast}\dashv (-)_{\ast}$, and then apply the universality of the $\delta$-functor $u^{\ast} \text{R}^{\bullet}f_{\ast}$ to get the desired extension.
Anyway, your feeling that you don't use anything special about quasi-coherent sheaves is completely right.
