I have problem connection my Mac Book Pro to Studio Display

Are you trying to use a Studio Display as a monitor for a MacBook Pro? That's often possible, but depends upon the specific model of Studio Display and the specific model of MacBook Pro.

Or trying to use a MacBook Pro as a monitor for a Mac Studio computer? That won't work. As Mr. Bennet-Alder indicated, that computer has video OUTPUT, but no video INPUTS.

The MacBook Pro (Retina, 13-inch, Early 2015) has two Thunderbolt 2 ports (with Mini DisplayPort connectors) and a HDMI port.

MacBook Pro (Retina, 13-inch, Early 2015) - Technical Specifications

The 27" 5K Studio Display (2022) takes Thunderbolt 3 input and has three downstream USB-C ports that carry USB data to downstream peripherals.

Studio Display - Technical Specifications

I don't know why you expected a USB-C to HDMI cable to work. Those are meant to be used with their USB-C end plugged into the video source (computer) and their HDMI end plugged into the TV or monitor – and not the other way around.

If you got an Apple Thunderbolt 3-to-2 adapter and a Thunderbolt 1/2 cable, you could connect one of the TB2 ports on the MBP to the Thunderbolt 3 port on the Studio Display (that's the one with the lightning bolt symbol). You might get a picture, but I'm not sure that you would. Apple's compatibility list for the 2022 Studio Display does not list the Early 2015 13" MacBook Pro. Your computer has a maximum resolution of 3840x2160 for TB2 video output, so at best you'd get 3840x2160's worth of pixels centered (shrunk) or scaled (ugly?) on that 5K display. At worst, even with a Thunderbolt connection that provided some connectivity between the MBP and the Studio Display, there would be something that kept it from working, and you'd be out the money which you spent on the TB 3-to-2 adapter and the TB 1/2 cable.

You're probably better off getting a third-party 4K (3840x2160 pixel) monitor for that MBP, and connecting the monitor via a mDP-to-mDP, mDP-to-DP, or HDMI-to-HDMI cable.

Apple Thunderbolt Display A1407 features a lead-in cable that has Thunderbolt-2 and MagSafe connections, ONLY.

(its twin, the Apple Cinema Display, has a lead in cord with Mini Display port (which looks identical to Thunderbolt-2) and includes a USB-2 connector as well as MagSafe)

• To use a Thunderbolt-2 Display, just plug it in to the ThunderBolt-2 port on your MacBook Pro 2015 model. No adapter is required.

• To use a Mini DisplayPort display, just plug it into the Thunderbolt-2 port on your MacBook Pro 2015 model. it auto-convert to Mini DisplayPort.

EDIT: 2015 model already has Thunderbolt-2 ports, so no adapters needed.

The 27-in 5K Apple Studio Display released in 2022 for US$1600, is beyond the ability of a 13-in 2015 MacBook Pro to support at full resolution.

That MacBook Pro model can only supply up to 3840 by 2160.

RE: "Apple Thunderbolt Display A1407 features a lead-in cable that has Thunderbolt-2 and MagSafe connections, ONLY."

Apple made two 27" monitors that require Thunderbolt input.

The 27" Thunderbolt Display (2011 ) was a 2560x1440 pixel monitor with downstream USB 2.0, Firewire 800, and Gigabit Ethernet ports. It connected using Thunderbolt 1.

The 27" Studio Display (2022) is a 5120x2880 pixel monitor with a Thunderbolt 3 input port, and downstream USB-C (USB) ports. MacTracker shows other, much older Studio Displays, ranging in size from 17" to 21".

I tend to think he has the 2022 Studio Display.

There's no place he could have plugged in the USB-C end of a USB-C to HDMI cable on a 13-inch Late 2015 MBP, on a 27" Thunderbolt Display, or on on the ancient 17" – 21" Studio Displays. There are places on the 2022 Studio Display where the USB-C plug would have physically fit.

Which MacBook Pro?

Which Studio Display? The 27" 5K one released in 2022?

How are you connecting them?

The MacBook Pro is a specialized portable computer. it is NOT a display, and has no display INPUTS.

If you need a display, use any generally available display or your TV set.

The current 2022 Apple Studio Display costs more than the current base model MacBook Pro 13-in.

That makes me doubt its use as a "found" display.

which display do you have?

OR what connector(s) on the lead-in cord?