Tow vehicle charging trailer battery

 I have a question I am planning on connecting our AGM tow vehicle battery to our trailer deep cycle wet cell battery. I plan on running wires directly from one battery to the other to charge the trailer battery while towing. Our tow vehicle only has a flat four pin connector so the tow vehicle only operates the trailer lights. My question will there be a problem having the two different kind of batteries. My internet research if I am understanding correctly there could be a problem but I have found that asking AI a question isn't always correct. They will only be connected together while towing and disconnected while not towing. But what I have read is that our tow vehicle will charge the AGM battery differently and the deep cycle trailer battery could be over charged. I am wondering if that is correct and there will be a problem. Thank you