The first Ext-groups between Schur functors in the category of modules over the $\Bbbk$-linearization $\Bbbk\mathbf{ gr}^{ \operatorname{ op}}$ of the opposite of the category of finitely generated free groups are computed for a filed $\Bbbk$ of characteristic $0$. The $\Bbbk$-linear category $\mathbf{ A}$ of Jacobi diagrams in handlebodies, which was introduced by Habiro and Massuyeau, has an $\mathbb{ N}$-grading whose degree $0$ part identifies with the category $\Bbbk\mathbf{ gr}^{ \operatorname{ op}}$. We compute the first Ext-groups in the category of $\mathbf{ A}$-modules between simple $\mathbf{ A}$-modules which are induced by Schur functors.
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