Lie supermodules are graded enveloping-superalgebra modules

With pointwise addition and scalar multiplication, $\mathrm{End}(V)$ is a complex vector space: sums and scalar multiples of linear maps are linear, and its vector-space laws hold pointwise in $V$. Write $\mathrm{End}(V{)}_{\bar{0}}$ for the endomorphisms preserving $V_{\bar{0}},V_{\bar{1}}$ and $\mathrm{End}(V{)}_{\bar{1}}$ for those interchanging them. If $p_i$ is the projection onto $V_{\bar{i}}$, every $T\in \mathrm{End}(V)$ decomposes uniquely as $T_0+T_1$, where $T_0=p_{0}T{p}_0+p_{1}T{p}_1$ and $T_1=p_{1}T{p}_0+p_{0}T{p}_1$; hence these spaces give a direct-sum grading. Composition sends parities $i,j$ to $i+j$ because an operator of parity $j$ maps $V_{\bar{k}}$ into $V_{\overline{k+j}}$, and the identity is even. Composition is associative and complex-bilinear, so by [F4] this makes $\mathrm{End}(V)$ a unital associative superalgebra. Given a $\mathfrak{g}$-supermodule, each homogeneous $x$ acts as a parity-$\mathrm{\mid }x\mathrm{\mid }$ endomorphism $v\mapsto x\cdot v$. The module identity in [F1] says exactly that this map preserves the Lie bracket into the endomorphism supercommutator, so [F3] extends it uniquely to a unital graded algebra action of $U(\mathfrak{g})$.

Conversely, restrict a unital graded left $U(\mathfrak{g})$-module action along ${\iota }_{\mathfrak{g}}$. Applying the quotient relation [F2] to $v$ gives $[x,y]v=x(yv)-(-1{)}^{\mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }y\mathrm{\mid }}y(xv)$; grading and bilinearity are inherited from the module action. Thus the restriction is a $\mathfrak{g}$-supermodule.

Starting from a supermodule, the extension in [F3] restricts to the original action by its defining property. Starting from a $U(\mathfrak{g})$-module, the extension of its restriction agrees with the original action on every generator ${\iota }_{\mathfrak{g}}(x)$ and on the unit; these generate the quotient algebra by [F2], so the actions agree on all of $U(\mathfrak{g})$.

A parity-preserving linear map intertwines the $\mathfrak{g}$-actions exactly when it intertwines the restrictions of the $U(\mathfrak{g})$-actions: one direction is restriction, and the other follows on products of generators and then on finite linear combinations. The correspondence is therefore mutually inverse on objects and morphisms.