show/hide this revision's text 2 Updated question based on a comment of the OP

We know that the evaluation functor $\Gamma(u, -):\Qco(X) ):Qcoh(X) \to {\cal O}_X(u)$ O}_X(u)-mod$ is a left exat exact functor preserving limits. We also know that the category of all quasi-coherent sheaves on scheme is locally finitely presented. SO So we may use "Adjoint Functor TheormTheorem" and deduce that there is a left adjoint for the evaluation functor.

Is there any explicit description for this adjoint?

The answer is true if we replace $\Qco(X)$ Qcoh(X)$ by the category of sheaves on X $X$ (see section 2 of "Relative Homological Algebra in Categories of Representations of Infinite Quivers" by S. Estrada)
show/hide this revision's text 1

A left adjoint for the evaluation functor \Gamma(, U)

We know that the evaluation functor $\Gamma(u, -):\Qco(X) \to {\cal O}_X(u)$ is a left exat functor preserving limits. We also know that the category of all quasi-coherent sheaves on scheme is locally finitely presented. SO we may use "Adjoint Functor Theorm" and deduce that there is a left adjoint for the evaluation functor. Is there any explicit description for this adjoint? The answer is true if we replace $\Qco(X)$ by the category of sheaves on X (see section 2 of "Relative Homological Algebra in Categories of Representations of Infinite Quivers" by S. Estrada)