By non-arbitrage, you buy the stock and hold it to the delivery date of the forward, only cost of funding (of cash) and equity dividend would be involved in the equity forward calculation. Where does cost of borrow come into play?
In finance just in general, you always assume you have 0 on day 1. So if I want to replicate a long fwd equity:
I am hypothetically saving me borrowing $x and paying an interest rate (where I collateralise my borrow with the equity). At the same time I am giving up the div on the eq:
Forward price = spot price + interest - dividend