Balanced Network max flow review

I would like a review for my algorithm for the following question:
A flow network is a “balanced network” if for every $$v∈V- {s,t}$$ it holds that $$c_in (v)=c_out (v).$$ Let G be a balanced network in which in-degree(s)=out-degree(t)=0
Find max-flow for G. My Algorithm so far:

1. We will define the following capacity-function: $$∀(u,v)∈E,f(ⅇ)=c(ⅇ)$$
2. Run over the adj. list and update for every edge it to be our function, and return c(e)

I was also wondering how to prove its a max flow in the correctness part Thank you

1. The flow defined by you, satisfies both the constraints: a) conservation of flow at each vertex -- by the definition of balanced network and b) the capacity constraint because $$f_{e} = c_{e}$$ for every $$e \in E$$. Therefore, it is a valid flow.
2. It is a max-flow because it is sending the flow of value which is equal to the capacity of the cut $$C$$ defined as: $$\{s\}$$ and $$V \setminus \{s\}$$, i.e. max-flow = $$\sum_{(s,u) \in E} c(u,v)$$ since there is no incoming edge to $$s$$. Note that any flow can not exceed the capacity of this cut. This fact follows from the min-cut max-flow theorem. i.e., max-flow $$=$$ min capacity of an $$s$$-$$t$$ cut $$\leq$$ capacity of cut $$C$$. Therefore, what we have is a max-flow.