How do we introduce our cats?

[Brunocatmon]

Dd and her family ( partner and 2 x dc 5 and 2 ) has moved from overseas to live with me until they can buy their own home ( maybe up to a year ).

She has brought her 1yr old cat, male. I have a 4 yr old male.

Its day 3. Both are neutered.

So far we've not let them loose near each other but as of last night we've let them look at each other whilst we're holding them. If my cat goes outside, as he does a lot. Cat 2 is allowed to roam the house. If my cat wants to come in and eat, cat 2 is contained in a room for a short period of time.

We're feeding them separately but in the same area, both are eating as usual.

My cat is sniffing around but isn't bothered by the new smell.

In the garden today we had my cat loose and cat 2 on a lead. I was sat with my cat giving treats and talking to him. Every so often he'd give a little growl towards cat 2 but was relaxed, no hackles up, happy to eat.

Are we doing the right thing? Never done this before.

I've been around cats my entire life, and the one thing I can say, every cat is an individual, every cat's personality and often background and experiences are different.
It makes sense to keep them under your watchful control whilst letting them see each other, smell one another, and size one another up.
Two males, even neutered males, can have confrontations, to determine dominance.
Some never get along, they can coexist , barely tolerating each other, whilst others will be more accepting about it. I've experienced both of those situations.
I often Google credible sites and take ideas and advice and as I need it.
I always think it helps to have a treats in your pocket to sweeten the mood if you think things might get tense.
I do hope that the two cats will lean toward tolerating each other; it makes for such a peaceful home!

Thank you, this is so stressful 🤣🤣

Yes I understand, it's a stressful situation.!
It's stressful for you all, and for the cats as well.
Both cat's lives have been upended,.
Cat2 has endured the stress of travelling, now finding itself in new surroundings with new people, possibly new food and a new cat.
Your cat has gone from being number one to having to share its living space with another.
It sounds to me that you and your DD are not rushing things and are doing the right things.
I completely understand how you feel on edge, as much as the cats do. I think it's key not to rush things. Both cats will settle down and find their quiet place, plenty of attention for them both to minimise jealousy, treats, and play. One of those toys on a stick I've found helpful to engage them both in play at the same time, play is a good outlet for pent-up stress. I hope everyday gets better for your family and your new cat family OP,. It does take time and patience, and it sounds as if you are doing a very good job so far!