From proposition $8.1.3$ of "Terrence Napier, Mohan Ramachandran-An Introduction to Riemann Surfaces".
Let $V$ and $W$ be real vector spaces.
$(a)$ The set $V⊕V$, together with the standard direct sum vector addition and
with scalar multiplication given by $z · (u,v) = (xu − yv,yu + xv)$ for all
$(u,v) ∈ V ⊕ V$ and $z = x + iy$ with $x,y ∈ R$, is a complex vector space $V_{\mathbb{C}}$
of dimension $\dim_{\mathbb{C}} V = \dim_{\mathbb{R}} V$. Denoting the element $(u,v) ∈ V_{\mathbb{C}}$ by $u + iv$
for all $u,v ∈ V$, we have $z · (u + iv) = (xu − yv) + i(yu + xv)$ for all
$z = x + iy$ with $x,y ∈ \mathbb{R}$. We also have a real linear inclusion $V \rightarrow V_{\mathbb{C}}$ given
by $v \mapsto v + i0 ↔ (v,0)$.
$(b)$ The map $V_{\mathbb{C}} → V_{\mathbb{C}}$ given by $u + iv \mapsto \overline{u+iv} ≡ u − iv$ (i.e., $(u,v) \mapsto(u,−v)$)
is a conjugate linear isomorphism.
$(c)$ Any real basis for $V$ is a complex basis for $V_{\mathbb{C}}$.
$(d)$ If $α,β ∈ \hom(V,W)$ are two real linear maps, then the complex linear exten-
sion of the map $λ = α + iβ$, i.e., the map $λ: V_{\mathbb{C}} → W_{\mathbb{C}}$ (which, in an abuse of
notation, we give the same name) given by $λ(u + iv) = α(u) − β(v) + i(β(u) +
α(v)) ∈ W_{\mathbb{C}}$ for all $u,v ∈ V$, is a complex linear mapping. Moreover, setting $\overlineλ ≡
α − iβ = α + i(−β)$, we get $\overline{λ(w)} = \overline{λ}(\overline{w})$ for each $w ∈ V_{\mathbb{C}}$. The above correspondence gives a (canonical) isomorphism $[\hom(V,W)]_{\mathbb{C}} \cong \hom(V_{\mathbb{C}},W_{\mathbb{C}})$.
In particular, we have $(V^∗)_{\mathbb{C}} \cong (V_{\mathbb{C}})^∗$. Furthermore, $α ∈ \hom(V,W)$ is injective (surjective) if and only if the associated complex linear extension is injective
(respectively, surjective).
The entire proposition is clear up to point $(d)$, although the definition of the complex linear extension is clear, I'm unsure of what is being stated about the conjugate and how does this give us an isomorphism between the complexification of the $\hom(V,W)$ space and the space $\hom(V_{\mathbb{C}},W_{\mathbb{C}})$. When writing $\overline\lambda=\alpha-i\beta$ what we mean is, take $λ(u + iv) = α(u) − β(v) + i(β(u) + α(v))$, denote its real part $\alpha$ and its imaginary part $\beta$ (sorry for the confusing notation, using the same symbol for different things but I'm trying to make sense of this very thing), and then take the conjugate as one would expect, or do we mean something else? Like take the conjugate of $\lambda$ and then extend it using the previous definition? Lastly, what is the supposed isomorphism we obtain from the above relationship on conjugates?