Beginning reta is not the same thing as switching over from tirz. Jumping straight in at 4mg is likely a poor choice, whereas a swap carries far less risk of trouble. When I last proposed simply changing over at 6-8mg, a number of people pushed back quite firmly, a few of them with fairly solid science on their side, and on the whole reta produces more side effects than tirz.
A great deal also hinges on which side effects you are currently getting from 15mg of tirz and how things went as your doses went up. The more side effects tirz gave you, the more slowly and cautiously you ought to move. Broadly speaking, the slower and more careful your approach, the lower the chance of suddenly hitting severe side effects, though it probably makes little difference to how many side effects you get in total at any given dose.
So you can either bring reta in gradually while tapering tirz down, or simply stop and swap. I do not see it as a major issue either way.
It also comes down to how much of a problem it is for you to sit at a lower reta dose for a while during the changeover, and whether putting some weight back on during that stretch would be a serious concern.
On a straight swap I would not begin above 4mg. Generally, if you want to move doses up faster than roughly every 4 weeks, the lowest-risk route is smaller doses given more frequently, so every 2 or 3 days or twice a week. That way each small added dose will either produce some side effects or not, which helps you judge whether pushing doses higher is going to make things worse. And usually, if you overshoot your tolerance, blood levels should fall back to pre-last-dose levels within only a few days. So something along the lines of 2mg 2 x a week, raising by 0.5mg per dose every week or 2, assuming no side effects show up. If it starts to take effect, with things like appetite suppression or side effects such as nausea, then you need to slow the increases right down. An approach like that would land you at 6-8mg within 2 to 4 weeks, with a fairly low risk of unexpected severe side effects. Have a look at glp plotter so you can see what different doses do to blood levels over time; you need some grasp of what is happening if you plan to deviate from the standard schedule. Also, do not do this if you have never been on a glp drug before. Once you reach a dose you are happy with and want to move to weekly, just step one dose up and the other one down gradually.
When I went from ozempic to tirz, I simply added an extra mg every day or two for a while, until it seemed to be suppressing appetite somewhat or causing mild nausea, but it got me to 15mg a week in about a month with no real problems.