If you want to claim the Migration Cape in Minecraft, you will need to first migrate your Java account to Microsoft. Once migrated, you will be able to equip your new cape via the option to do so under your profile on the Mojang website.
That said, to migrate your Minecraft Java account to Microsoft, you will first need to log in to your Mojang account.
Players are able to log in to their Mojang accounts by heading over to the official Mojang website and punching all of their info into the email and password options. You will need to use the email you created your java account with, along with your password.
Once logged in, head to your profile and find a button that says Move My Account to begin the process of migrating your account to Microsoft.
First, you will need to confirm your email address. To do this will require a code Mojang will send to the email you created the account on. Grab the code and input it into the box that will appear right after you get the code.
Next, you will need to create both a Microsoft and Xbox account. If you have each already, just log in to the Microsoft account now, and your Java account will automatically migrate over and be linked.
If you don't have one yet, create one now for both Microsoft and Xbox, and log in with the Microsoft account to migrate your java account to Microsoft.
Once your account is migrated, head back to your profile and you should see a Change Cape option with the Migration Cape under the list of options. Change the cape to the Migration Cape, and then log in to the Minecraft launcher using Microsoft by going back to the login page.
Hit Login with Microsoft and punch all of your Microsoft info to log in. Afterward, you will have the cape permanently until you remove it from your character in Minecraft.
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