Should I transfer cash from one brokerage to another?

I have an account in E-Trade and one in Fidelity. The fidelity account was opened with an employer for stock grants. I sold some of the stock recently and wanted to reinvest some of the cash.

My original plan was to transfer the cash to E-Trade and add it to my mutual fund account. Should I stick with this plan or just invest within Fidelity itself? I'm only coming here because I was trying to make the transfer and when I clicked the "Transfer terms and conditions" on E-Trade it was a broken link. I can't read the damn ToS for the transaction…

I was originally thinking one location for everything would be easier, but the transfer seems to make that less so. Should I keep my brokerages diversified as well?